Tell us what is in calc III and I can perhaps help you.
I personally think that to undertake a study of PDEs you should know the following:
1) ODEs
2) Partial differentiation including vector calculus
3) Fourier series/transforms
4) Laplace transforms

The obvious general answer is that the partial differential operators laplacian, curl, divergence, and gradient themselves appear in actual PDEs. The easiest example would be to point to the role of the laplacian in the laplace, heat, and wave equations. Taking this example further, Stokes theorem, Green's identities, and in general vector calc identities are used to establish properties of the solutions to the aforementioned PDEs such as uniqueness, regularity, sign of eigenvalue, just to name a few.

The gradient appears (somewhat implicitly) often early on in basic PDEs such as [itex]u_x + u_y = u[/itex], where the method of characteristics is commonly used. Curl and divergence appear, for example, in Maxwell's equations and many nonlinear equations.

I recently asked the professor who will be teaching my PDE class this upcoming semester what I need to review, and she said: " What is really needed
is review of solving ODE's via separation of variables, integrating factor,
and solution of second-order constant coefficients homogeneous solutions."